Theorem 3eqtr3a

Description: A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015)

	Ref	Expression
Hypotheses	3eqtr3a.1	⊢ 𝐴 = 𝐵
	3eqtr3a.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
	3eqtr3a.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion	3eqtr3a	⊢ ( 𝜑 → 𝐶 = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3eqtr3a.1	⊢ 𝐴 = 𝐵
2		3eqtr3a.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		3eqtr3a.3	⊢ ( 𝜑 → 𝐵 = 𝐷 )
4	1 3	syl5eq	⊢ ( 𝜑 → 𝐴 = 𝐷 )
5	2 4	eqtr3d	⊢ ( 𝜑 → 𝐶 = 𝐷 )